316 Mr. Forsyth. Linear Differential Equations. [Jan. 12, 



Up to this point the results in the memoir which relate to the 

 derivation of covariantive forms have been synthetically obtained ; the 

 eighth (and last) section relates to their analytical derivation. It is 

 shown that, for a nomographic transformation of the independent 

 variable applied concurrently with the proper transformation of the 

 dependent variable, the canonical form of the differential equation is 

 maintained. These transformations are applied to prove, by the 

 method of infinitesimal variation, that every concomitant in its 

 canonical form satisfied the linear partial differential equation 



m-n-i r rfrfy \ 



p = n-\ r = <r-\ r J a. n 



+ A *. [^-iO-+i}V->4J 



7b ! 



where rr = — — r . This is called the form-equation. Such a con- 



p\n— p\ J 1 



comitant also satisfies the equation 



p-n-\ t = t-\ r- rich ~\ 



where \ is the index of the concomitant. This is called the index 

 equation ; and, when the form of is known, it merely determines \, 

 which can be written down from an inspection of the concomitant. 



These equations are applied, (i) to the identical covariants in u, — 

 (ii) to the invariants derived from 3 , — for each of which simplified 

 equivalent functions are obtained for derivatives of order higher than 

 the third, — and (iii) to verify that the Jacobian of a priminvariant and 

 any of its derived invariants satisfies the equations. Lastly, by means 

 of the theory of partial differential equations, it is proved that the 

 aggregate of concomitants obtained in the earlier part of the memoir 

 is complete, that is, that any concomitant can be expressed as an 

 algebraical function of the members of that aggregate. 



